A228336 Triangle read by rows: the Z-transformation of the Catalan triangle A033184.

1, 1, 1, 2, 2, 1, 4, 6, 3, 1, 10, 15, 12, 4, 1, 25, 45, 36, 20, 5, 1, 70, 126, 126, 70, 30, 6, 1, 196, 392, 392, 280, 120, 42, 7, 1, 588, 1176, 1344, 960, 540, 189, 56, 8, 1, 1764, 3780, 4320, 3600, 2025, 945, 280, 72, 9, 1, 5544, 11880, 14850, 12375, 8250, 3850, 1540, 396, 90, 10, 1

Offset

0,4

Links

Table of n, a(n) for n=0..65.

Yidong Sun and Fei Ma, Four transformations on the Catalan triangle, arXiv preprint arXiv:1305.2017 [math.CO], 2013.

Yidong Sun and Fei Ma, Some new binomial sums related to the Catalan triangle, Electronic Journal of Combinatorics 21(1) (2014), #P1.33.

Example

Triangle begins:
   1;
   1,   1;
   2,   2,   1;
   4,   6,   3,  1;
  10,  15,  12,  4,  1;
  25,  45,  36, 20,  5, 1;
  70, 126, 126, 70, 30, 6, 1;
  ...

Mathematica

c[n_, k_] := Boole[k <= n] Binomial[2n - k, n] (k + 1)/(n + 1);
T[n_, k_] := Module[{nn, kk}, If[OddQ[n], nn = (n + 1)/2, nn = n/2]; If[OddQ[k], kk = (k - 1)/2, kk = k/2]; If [OddQ[n], If[OddQ[k], c[nn + kk, 2kk + 1] c[nn + kk + 1, 2kk + 2], c[nn + kk, 2kk] c[nn + kk, 2kk + 1]], If[OddQ[k], c[nn + kk + 1, 2kk + 1] c[nn + kk + 1, 2kk + 2], c[nn + kk, 2kk] c[nn + kk + 1, 2kk + 1]]]];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Oct 04 2018, from PARI *)

Prog

(PARI) C(n, k) = (k<=n)*binomial(2*n-k, n)*(k+1)/(n+1);
T(n, k) = {my(nn, kk); if (n % 2, nn = (n+1)/2, nn = n/2); if (k % 2, kk = (k-1)/2, kk = k/2); if ((n % 2), if (k % 2, C(nn+kk, 2*kk+1)*C(nn+kk+1, 2*kk+2), C(nn+kk, 2*kk)*C(nn+kk, 2*kk+1)), if (k % 2, C(nn+kk+1, 2*kk+1)*C(nn+kk+1, 2*kk+2), C(nn+kk, 2*kk)*C(nn+kk+1, 2*kk+1))); } \\ Michel Marcus, Feb 13 2014

Crossrefs

Cf. A033184, A228334, A228335, A228337.

Sequence in context: A175136 A091869 A112307 * A111062 A369632 A363493

Adjacent sequences: A228333 A228334 A228335 * A228337 A228338 A228339

Keyword

nonn,tabl

Author

N. J. A. Sloane, Aug 26 2013

Extensions

More terms from Michel Marcus, Feb 13 2014

A-number for Catalan triangle changed by Michel Marcus, Feb 13 2014

Status

approved